Conventional digital signal-processing techniques consider signals that are represented by a set of uniformly spaced sample values. Although most processing algorithms are derived within a purely discrete framework, there are a variety of biomedical problems (detection of anatomical structures in images, registration) that would be better formulated by considering a signal as a continuous real-valued function defined over some domain. This study is concerned with the development of such new processing techniques that represent signals and images by continuous polynomial spline functions. What makes this type of approach feasible is the availability of fast algorithms for spline interpolation and approximation that we developed previously. The spline formulation provides a sound mathematical foundation for this type of problem, and typically results in algorithms that outperform the best conventional approaches. We have shown that these techniques are especially appropriate for performing geometrical transformations of images (scaling, rotation, affine transformations). Our new high-quality spline-based methods have been found to be extremely useful for the reslicing of volumetric data (PET, MRI, and fMRI). They have also resulted in the design of fast algorithms for the wavelet transform.